Notice that it follows from [[1]}, Lemma 3.4] that for any geometric point \(s\)  of \(S\)  and any subpair \((\mathcal {F}_{s}^\prime ,\varphi _{s}^\prime )\)  of \((\mathcal {F}_{s},\varphi _{s})\)  with \(r(F_{\mathcal {E}_{s}}(\mathcal {F}^\prime _{s}))=r^\prime _{s}\)  satisfying \(0<r^\prime _{s}<r\) , we have
\(\frac{h^0(X_{s},F_{\mathcal {E}_{s}}(\mathcal {F}^\prime _{s})(m))}{r_{s}^\prime }&\le &\frac{r_{s}^\prime -1}{r_{s}^\prime }\left[\binom{\hat{\mu }_{\mathrm {max}}(F_{\mathcal {E}_{s}}(\mathcal {F}^\prime _{s}))+m+C_{s}}{d}\right]_{+}+\frac{1}{r_{s}^\prime }\left[\binom{\hat{\mu }(F_{\mathcal {E}_{s}}(\mathcal {F}^\prime _{s}))+m+C_{s}}{d}\right]_{+}\\&\le &\frac{r-1}{r}\left[\binom{\hat{\mu }_{\mathrm {max}}(F_{\mathcal {E}_{s}}(\mathcal {F}^\prime _{s}))+m+C}{d}\right]_{+}+\frac{1}{r}\left[\binom{\hat{\mu }(F_{\mathcal {E}_{s}}(\mathcal {F}^\prime _{s}))+m+C}{d}\right]_{+}\) 
where \(C_{s}:=\widetilde{m}\deg (\mathcal {O}_{X_{s}}(1))+r_{s}^{\prime 2}+f(r_{s}^\prime )+\frac{d-1}{2}\)  and \(C:=\widetilde{m}\deg (\mathcal {O}_{X_{\bar{s}}}(1))+r^2+f(r)+\frac{d-1}{2}\) . Here, \([x]_{+}:=\max \lbrace 0,x\rbrace \) , \(f(r)=-1+\sum _{i=1}^r\frac{1}{i}\) , and \(\bar{s}\) , \(\widetilde{m}\)  are chosen as in the proof of Proposition REF . Now the proof follows from the similar argument in [[1]}, Lemma 3.5] by applying Proposition REF , Grothendieck Lemma REF  which is used for two set-theoretic families restricted on the associated closed subschemes of \(S\)  (where the flatness of \(\mathcal {F}\)  is preserved by base change) induced respectively by bounding the slope of \(F_{\mathcal {E}_{s}}(\mathcal {F}_{s}^\prime )\)  and \(F_{\mathcal {E}_{s}}(\mathcal {G}_{s})\)  (e.g., as in Case \((a)\)  and Case \((\tilde{b})\)  in the proof of [[1]}, Lemma 3.5]), and Kleiman criterion for stacks in [[4]}, Theorem 4.12].
